Ground state solitary waves local controllability for the nonlinear focusing Schrödinger equation in the mass critical and slighlty mass subcritical regime

In this paper, we prove the exact local controllability around different ground state solitary wave for the slightly subcritical mass and mass critical nonlinear Schrödinger equation. More precisely, if Q c1 and Q c2 denotes the ground states with two different scaling, we prove the exact local controllability from Q c1 to Q c2 in a minimal time depending on c 1 and c 2. The results presented relies on the blow-up profile in the mass slightly supercritical case and mass critical case